Question

A right triangle has sides 3, 4, and 5. A smaller triangle is drawn inside it with its vertices on the sides of the larger triangle, such that it is similar to the larger triangle. If its perimeter is 6, what is its area?

• A. 1.5
• B. 2.4
• C. 3
• D. 6

Answer 1

Answer:

A. 1.5

Since the inner triangle is similar to the 3–4–5 triangle:

Perimeters ratio = similarity ratio

Original triangle perimeter:
3 + 4 + 5 = 12

Smaller triangle perimeter = 6

So similarity scale factor:
$k = \frac{6}{12} = \frac{1}{2}$

Scale the sides

Original sides: 3, 4, 5
Smaller sides:
$\frac{3}{2}, \frac{4}{2}, \frac{5}{2} = 1.5, 2, 2.5$

Area scaling rule

Area scales by square of similarity factor:

Original area:
$\frac{1}{2} \times 3 \times 4 = 6$

Smaller area:
$6 \times \left(\frac{1}{2}\right)^2 = 6 \times \frac{1}{4} = 1.5$

Final Answer:

$\boxed{1.5 \text{ square units}}$